Defining Slope

2 MIN READ · PRACTISE AT YOUR OWN PACE · FREE

Idea

Slope is one number that tells you two things about a line: how steep it is, and which way it leans.

Think of walking up a hill. The slope compares how far you climb to how far you walk forward.

Rule

Slope is the ratio of vertical change to horizontal change between any two points on a line:

m = \frac{\text{rise}}{\text{run}}

Read it as: for every unit you move right, the line moves $m$ units up (or down, if $m$ is negative).

How it is used

Two separate pieces of information live inside $m$ :

Magnitude = steepness. $|m| = 3$ climbs harder than $|m| = \dfrac{1}{2}$ . Bigger absolute value, steeper line.
Sign = direction. Rising left-to-right means $m > 0$ ; falling means $m < 0$ . Here is the key fact: a straight line has the same slope everywhere.

Pick any two points on it and draw the right triangle. Pick another two and draw a bigger one. The triangles have different sizes, but $\dfrac{\text{rise}}{\text{run}}$ gives the same number every time. This is what makes a line straight.

Worked example

A line passes through $(0, 0)$ and $(4, 2)$ . Check its slope with two different measurements. Using the nearby point: rise $= 2$ , run $= 4$ , so $m = \dfrac{2}{4} = \dfrac{1}{2}$ . Now measure to the farther point $(8, 4)$ on the same line: rise $= 4$ , run $= 8$ , so $m = \dfrac{4}{8} = \dfrac{1}{2}$ . Same ratio - the triangles are similar, the slope is constant.

Explore

Drag either point and watch the rise and the run. The slope is one divided by the other. Make the line steeper and the number grows. Tilt it the other way and the number turns negative.

Rise over run

Try this. Give A and B the same y for zero slope, then the same x for undefined slope. Coincident points do not determine a slope.

Point A · x-2Point A · y3Point B · x3Point B · y1

A = (-2, 3), B = (3, 1) · Rise = -2 · Run = 5 · Slope = -0.4

PAUSE & THINKA quick check, not a grade

Try it yourself.

Choose an answer and explain your reasoning to yourself. Use a hint if you get stuck.

A line rises 6 units while moving 3 units right. What is its slope?

$1/2$

$3$

$2$

Hint 1 · Find a starting point

Slope compares rise to run in that order.

Hint 2 · Take the next step

Divide vertical change by horizontal change: 6/3.

Show the reasoning

Answer: $2$

The slope is 2: every unit of horizontal movement corresponds to a rise of 2 units. A ratio carries more information than either change alone.

Before moving on: what would make one of the other answers wrong? Saying why is part of understanding.

MAKE IT YOURS

Pause before the next idea.

Can you explain how the visualization connects to this lesson’s goal? If a step still feels uncertain, put this lesson on your review list and try the check again another day.

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